i know everyone plays with their own rules, but from my perspective, a square root spends a 2 - it's not an operator like + or ! . there's an implicit "2" since you could also take the cube root or fourth root

It’s true cheating, but if you assume the Peano axioms, you can write any integer with one zero without having to resort to complex functions such as √. For example (if I didn’t miscount the S-es or parentheses),

  S(S(S(S(S(S(S(0)))))))) = 7

there is a related problem (inspired by an interview questions from ITA software, a part of Google now) that asks to find the smallest positive integer that cannot be expressed using a digit K (in [2,9]) N times in an expression that involves the four basic arithmetic operations and the parentheses: +, -, *, /, (, and ).

to make the problem more interesting, the list of operations can be extended to include exponentiation and concatenation.

for example, using the four basic operations for N=4, and K in [2,9], we get:

7 cannot be computed in 4 2s; 11 cannot be computed in 4 3s; 10 cannot be computed in 4 4s; 8 cannot be computed in 4 5s; 9 cannot be computed in 4 6s; 4 cannot be computed in 4 7s; 5 cannot be computed in 4 8s; 4 cannot be computed in 4 9s;

Popular last year (368 points, 165 comments) https://news.ycombinator.com/item?id=43145753

22 is cheating because 22 = 2*10+2 is arbitrary in base 10. If you write 22 in some base, you have to write the base, and that's an extra number.

Never understood why 7 is considered difficult...

7 = ⌈ 2.2² + 2 ⌉

I'll just drop this here (yes, it's 4 instead of 2 - they're twice as good): https://fourmula.awsum.info/

Spoilsport.

And JFYI since December is near: https://rant.gulbrandsen.priv.no/hack/advent-calendar-for-ne...

There's a direct trivial solution for any natural number using the successor function S(a) = a + 1 from Peano arithmetic.

Encode 0 as 2 + 2 - 2 - 2 then apply S a times to that 0.

Only depends upon natural numbers & Peano stuff, no need for construction of rationals, irrationals, logarithms, etc.